Micro-Electrical-Mechanical Systems (MEMS) resonators have great potential for on-chip selective radio frequency (RF) applications such as oscillators and filters. One of the most important factors in MEMS resonator design is a quality factor (Q) of the MEMS resonator. The quality factor (Q) generally compares a resonant frequency of the MEMS resonator to a rate at which it dissipates energy. The frequency selectivity of the MEMS resonator is dependant on the quality factor (Q). The higher the quality factor (Q), the greater the frequency selectivity. Thus, a high quality factor (Q) is needed for applications in which frequency selectivity is required.
FIGS. 1 and 2 illustrate a conventional MEMS resonator 10 including a resonator body 12 connected to anchors 14 and 16 by support structures 18 and 20, respectively. The support structures 18 and 20 are more specifically support beams 18 and 20. The anchors 14 and 16 are either connected to or are part of a substrate 22. As illustrated in FIG. 2, the resonator body 12 and support beams 18 and 20 are separated from the substrate 22 by a gap 24 having some height (h). In the conventional MEMS resonator 10, the support beams 18 and 20 are each designed such that their lengths are exactly quarter-wavelength (λ/4), which is defined by the equation:
            λ      4        =                  1                  4          ·                      f            o                              ⁢                                    E            eff                                ρ            eff                                ,where fO is a resonant frequency in Hertz of the resonator body 12, Eeff is the Young's Modulus of the support beams 18 and 20, and ρeff is a density of the support beams 18 and 20.
However, the conventional MEMS resonator 10 having the quarter-wavelength (λ/4) support beams 18 and 20 has several issues that limit its quality factor (Q) or result in a reduction in its quality factor (Q). One issue with the conventional MEMS resonator 10 that limits or reduces the quality factor (Q) is that there is a significant amount of vibrational energy at the anchor points of the support beams 18 and 20 that is dissipated through the anchors 14 and 16 into the substrate 22 when the conventional MEMS resonator 10 is in vibration mode. Note that the points at which the support beams 18 and 20 are connected to the anchors 14 and 16 are referred to herein as anchor points. FIG. 3 presents a modal analysis of the conventional MEMS resonator 10. In general, the quarter-wavelength (λ/4) support beams 18 and 20 do not vibrate with mechanically symmetric mode. For the quarter-wavelength (λ/4) support beams 18 and 20, a boundary condition is created (i.e., the anchor points are fixed with no vibration) in order to provide vibration at the desired frequency. If all boundary conditions are removed, the quarter-wavelength (λ/4) support beams 18 and 20 will not vibrate at the desired frequency of the conventional MEMS resonator 10.
The quarter-wavelength (λ/4) support beams 18 and 20 have maximum stress at the anchor points and, therefore, the anchors 14 and 16 must provide counter action in order for there to be no movement at the anchor points. In reality, the anchors 14 and 16 close to anchoring points are floating together with the resonator body 12 and the quarter-wavelength (λ/4) support beams 18 and 20 because of manufacturing issues. As a result, near the anchor points, the anchors 14 and 16 vibrate with some displacement in order to counter act with quarter-wavelength (λ/4) movement on the quarter-wavelength (λ/4) support beams 18 and 20. As a result, a significant amount of vibrational energy is dissipated through the anchors 14 and 16 into the substrate 22. The dissipation of energy through the anchors 14 and 16 into the substrate 22 is another anchor loss that limits or reduces the quality factor (Q) of the conventional MEMS resonator 10.
Another issue with the conventional MEMS resonator 10 is that the quarter-wavelength (λ/4) support beams 18 and 20 are relatively long. For example, for a desired resonant frequency (fO), the length of the support beams 18 and 20 may be on the order of tens of microns. The relatively long support beams 18 and 20 raise fabrication issues and may further result in the resonator body 12 bending and touching the substrate 22, which would of course degrade the quality factor (Q) of the conventional MEMS resonator 10.
Thus, there is a need for an improved MEMS resonator that eliminates or reduces anchor losses, thereby improving a quality factor (Q) of the MEMS resonator.